In this talk we will discuss the quantitative analysis concerning 
the asymptotic behavior of time inhomogeneous Markov chains on a 
finite state space $V$.  We will study examples of 
time inhomogeneous Markov chains with a wave like behavior.  
The transition kernel of these chains at time $n$ 
is obtained by transporting a prescribed Markov kernel $K$ by the map 
$g^{n-1}$ where $g$ is a given bijection of $V$.
We show that this construction leads to interesting examples and  obtain
quantitative results for some of these examples.
This is joint work with Laurent Saloff-Coste.